{"id":"n-cfb1b41332b76420","revision":1,"first_index":1,"fingerprint":"f6-f4ff2f177f5d4ad6","name":"Triangle, T(n, k) = (1/2)*(n+2)! * [x^k]( p(x, n) ), where p(x,0) = 1, p(x,1) = -x, P(x, n) = (1/(n+1))*( (2*n-x)*P(x, n-1) - n*P(x, n-2) ), read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["-8","-16","4"],["-60","-65","50","-5"],["-384","-168","462","-108","6"],["-2380","763","3836","-1624","196","-7"],["-14208","21248","29560","-21472","4256","-320","8"],["-73836","302571","199998","-269127","78840","-9387","486","-9"],["-176000","3761240","854530","-3288940","1360150","-228880","18430","-700","10"],["3824964","44711623","-7017472","-39417422","22743644","-5098676","568568","-33242","968","-11"],["104573760","520004832","-304428396","-457688256","376007784","-108589488","15755712","-1261536","56184","-1296","12"],["1945371428","5886398167","-6815383510","-4941747993","6205006704","-2266280562","411844524","-42381690","2567292","-90181","1690","-13"],["32711977536","62684925464","-131346678218","-43493908108","102528132630","-47013916992","10438247148","-1326990504","102557532","-4877880","138782","-2156","14"],["531914802900","563326733715","-2397105814620","-107246569140","1692765146100","-977600263005","260842113720","-39885570720","3778113420","-228181575","8762820","-206220","2700","-15"],["8559039145216","2215892686592","-42772786258448","9096450753472","27705346994816","-20472159938048","6495090715824","-1172234760576","132411375744","-9752365568","474022736","-15023168","297472","-3328","16"],["137181098014260","-82110716803997","-754474965239434","362743328297221","441787844278696","-432698962140997","162267947502466","-34095196345323","4502693112852","-393357437583","23236844026","-929926537","24755536","-418319","4046","-17"],["2183015105955072","-3565531955112552","-13157883856122390","10469675504971188","6605590559327046","-9232652254914576","4085585584428474","-989434986010020","150528157151526","-15286751662248","1067296512942","-51785143956","1737731970","-39426912","575406","-4860","18"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["-20","-80","-192","784","19072","229536","2299840","20282944","144429312","385305856","-14720394752","-469430929920","-10222242357248","-195267955503104","-3465377681743872"],"nonn":false,"definitions":[{"position":0,"source":"oeis_catalog:A136532","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["-8","-16","4"],["-60","-65","50","-5"]],"recurrence":{"denominator":{"coeffs":[["1"],["1/2"]]},"terms":[{"coeff":{"coeffs":[["3","-3/2"],["4","-1/2"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3"],["-11/2"],["-3"],["-1/2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159262,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-8965e6e864ed875e"]},{"id":159263,"position":1,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"strict","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2","region_exact":null,"edges":["e-8965e6e864ed875e"]}],"region_exact":{},"incoming_edges":[{"id":"e-8965e6e864ed875e","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-cfb1b41332b76420","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"445c7a3367efdf2a476162edc93f1550deac0bdd78325d5dc45b25a0771c6e84","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}},{"claim":"interlacing","from_index":2,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[{"oeis_id":"A136532","relation":"tail"}],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:         -8,        -16,           4;\n   2:        -60,        -65,          50,          -5;\n   3:       -384,       -168,         462,        -108,          6;\n   4:      -2380,        763,        3836,       -1624,        196,          -7;\n   5:     -14208,      21248,       29560,      -21472,       4256,        -320,         8;\n   6:     -73836,     302571,      199998,     -269127,      78840,       -9387,       486,        -9;\n   7:    -176000,    3761240,      854530,    -3288940,    1360150,     -228880,     18430,      -700,      10;\n   8:    3824964,   44711623,    -7017472,   -39417422,   22743644,    -5098676,    568568,    -33242,     968,    -11;\n   9:  104573760,  520004832,  -304428396,  -457688256,  376007784,  -108589488,  15755712,  -1261536,   56184,  -1296,   12;\n  10: 1945371428, 5886398167, -6815383510, -4941747993, 6205006704, -2266280562, 411844524, -42381690, 2567292, -90181, 1690, -13;\n","recurrence_sentence":"The row polynomials satisfy ((1/2)*n+1)*P(n,x) = (n^2+4*n+3-((1/2)*n+3/2)*x)*P(n-1,x) - ((1/2)*n^3+3*n^2+(11/2)*n+3)*P(n-2,x) for n >= 3, with P(1,x) = -8 - 16*x + 4*x^2, P(2,x) = -60 - 65*x + 50*x^2 - 5*x^3.","mathematica":"P[1, x_] := -8 - 16*x + 4*x^2;\nP[2, x_] := -60 - 65*x + 50*x^2 - 5*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 4*n + 3 - ((1/2)*n + 3/2)*x)*P[n - 1, x] - ((1/2)*n^3 + 3*n^2 + (11/2)*n + 3)*P[n - 2, x])/((1/2)*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = -8 - 16*x + 4*x^2; v[2] = -60 - 65*x + 50*x^2 - 5*x^3; for(n = 3, N, my(i = n); v[i] = ((n^2 + 4*n + 3 - ((1/2)*n + 3/2)*x)*v[i - 1] - ((1/2)*n^3 + 3*n^2 + (11/2)*n + 3)*v[i - 2])/((1/2)*n + 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cfb1b41332b76420","steps":[],"edges":[],"text":"A136532"},"headline":"Triangle, T(n, k) = (1/2)*(n+2)! * [x^k]( p(x, n) ), where p(x,0) = 1, p(x,1) = -x, P(x, n) = (1/(n+1))*( (2*n-x)*P(x, n-1) - n*P(x, n-2) ), read by rows.","chain":null}